Discrete modulation instability and localized waves of the coupled semi-discrete Hirota equation

Abstract

In this paper, we use the numerical simulation of the discrete Hirota equation to point out the localized modes and wave patterns. The discrete modulation instability is used to demonstrate how the nonlinear term generates unstable or stable modes when a little perturbation is introduced. We have also stated that for strong enough nonlinear terms, new bands emerge. Furthermore, by applying an external periodic force, we were able to push one end of the discrete Hirota model into the forbidden frequency gap. We established the threshold amplitude expression and the driven amplitude. According to one finding, the threshold of supratransmission decreases as the nonlinear term of the discrete Hirota equation increases. For specific times of propagation, we showed the propagation of the train of waves as well as the modulated wave patterns. We demonstrated the wave train traveling through the band gap, where energy jumps from the bottom to the top and vice versa, over a significant period of time. These findings will almost certainly pave the way for new nonlinear dynamics applications.